A method and apparatus for validating labelled, ordered parse trees according to a context-free grammar specification

Abstract: Some document languages such as the Standard Generalized Markup Language (SGML) represent documents as trees (Fig. 1) with each node of the tree labelled with a tag and each node's immediate descendants taken in order having tags that satisfy a production corresponding to the parent's tag. Thus, a document is represented as a complete parse tree satisfying the production rules of a grammar. To simplify maintaining a valid document/parse tree at all stages, an efficient check is made whether a document is valid or can be extended to a valid document and furthermore to show how a document could be changed to be valid. A document can be extended to a valid document if is equal to a valid document with possibly some nodes deleted. External (leaf) nodes can just be deleted. Internal nodes are deleted by replacing the arc from the parent with multiple arcs (in the same order) from the parent to each of the children of the deleted node. A grammar is constructed from the given grammar that includes those documents that can be completed to a valid document and is called a gapped grammar. This invention teaches how a gapped grammar can be constructed from a grammar, and how a parser/checker is efficiently implemented.

Description: The present invention relates to methods for editing documents Documents can be described by internal computer structures in accordance with varying computer languages. For example, a document represented by one computer may use the Word Perfect® language while a document represented by a different computer may use the Microsoft Word® language. In order to make document processing more efficient, it is often desirable to enable interchange between documents represented using different languages. Thus, there should be consistency between interrelated documents. In order to provide such consistency enabling interchange of documents, the Standard Generalized Markup Language (SGML) for example, was developed as an external representation making document interchange possible. There are many editing applications wherein objects to be edited have hierarchical structures. Thus, any editing step must conform to a formal specification of possible content. The editing of documents described in SGML document type definitions is a specific instance of structured document editing. The article "syntax-directed editing of general data-structures", C. Fraser, ACM SIGPLAN Symposium on Text Manipulation, vol. 16, no. 6, 1981 describes a syntax-directed editor for data structure definitions. Structured document editing has been deemed by many users as an excessively restrictive way of constructing structured objects (e.g., programs or documents) because of the strong ordering constraints (i.e., top- bottom, and sometimes left-right, as well) that grammar-based editors typically impose. Thus a more relaxed version of syntax-directed editing is desirable. Often the class of hierarchical structures of interest can be made to correspond to incomplete parse trees of a particular context-free grammar (set of rules), i.e., any tree remaining after having deleted zero or more proper subtrees of a parse tree of the same grammar. For example, given the following grammar, it would be desirable to easily and efficiently determine whether a user could freely build a hierarchical structure; Doc → (Front, Body, Back) Body → (Intro, Mn. Sections + , App. + ) A parse tree corresponding to the above grammar is illustrated in Fig. 1. An SGML document class is defined by a document type definition or DTD (context-free grammar). In accordance with SGML, a document is classified as weakly valid if it can be made valid solely by the insertion of additional structural mark-up in appropriate places. For example, if "Front" was missing from the parse tree of Fig. 1 as shown in Fig. 2, the document would be classified as weakly valid since the simple insertion of "Front" would validate the parse tree. Similarly, if "Body" was missing from the parse tree of Fig. 1 as in Fig. 3, the document would also be classified as weakly valid since the insertion of begin and end tags around the [Intro, Sections + , Appendix] would result in a valid parse tree. A document is valid if and only if the structure is a member of language defined by the DTD. Thus, if, for example, a user wanted to add a node "x" beneath node "Body", it would be highly desirable to enable one to easily check for validity of this action, thus determining whether the action is legalized if some action can be taken. It is further desirable to perform such a check in an arbitrary, convenient order to enable the user to freely work on document segments. This invention is intended to test a sufficient condition for a tree to be embedded in a regular tree according to a grammar. In accordance with one aspect of the present invention, we provide a method of editing a document according to claim 1. In accordance with a second aspect of the present invention, a method of editing a document according to a computer language comprises determining whether a given parse tree generated from the document can be extended to a valid parse tree of a given context-free grammar G with only an insertion of additional nodes, by performing the steps of: constructing regular expression gap(X) for each non-terminal symbol X of the given context-free grammar G; checking each node having one of the non-terminal symbols of the given parse tree for child nodes whose names form a string in a language of the corresponding regular expression gap(X); and determining, when the names form a string in the language of the corresponding regular expression gap(X), that the given parse tree can be extended to the valid parse tree. A method in accordance with the present invention determines whether or not an SGML document is valid after at least one editing step. In addition, if the resulting document is weakly valid but not valid, then a possible completion of the document is indicated that will make the document valid. The document is checked to see whether components are thus far legal according to structure compatibility. If any component is determined to not be legal, the method looks at the point of failure and determines what could be done to render the document legal. By way of example only, methods in accordance with the invention will be described with reference to the accompanying drawings, wherein: Figure 1 (already described) is an exemplary parse tree corresponding to given grammar; Figure 2 (already described) illustrates an example of a horizontally gapped parse tree; Figure 3 (already described) illustrates an example of a vertically gapped parse tree; Figure 4 is a block diagram of an editing apparatus; Figure 5 is an exemplary parse tree corresponding to an example grammar; Figure 6 is an exemplary horizontally gapped parse tree corresponding to an example grammar; Figure 7 is an exemplary vertically gapped parse tree corresponding to an example grammar; Figure 8 is a reachability graph with strongly connected components and ranks indicated for an example grammar; and Figures 9 - 13 are exemplary finite state machines useful for weakly validating a parse tree for an example grammar. The use of document rules to form a document can be illustrated in the form of a parse tree, the parse tree having nodes labelled with symbols of the document rules (grammar productions). When the document is edited, the location of arcs and nodes of the original parse tree change. Methods in accordance with the present invention, which may be performed using the editing apparatus of Fig. 4, enable rapid validation of the newly created parse tree as described herein below. For a context-free grammar G = 〈V, Σ, S, P〉 , where V is the set of symbols (both terminal and non-terminal) of the grammar (nodes), Σ is the set of terminal symbols of the grammar, S is the start symbol of the grammar and P is the set of productions of the grammar with general regular expressions permitted for right-hand-sides of productions, T is assumed to be a parse tree of grammar G. A further assumption is made that the only productions of G having terminal symbols are of the form A → a, wherein a is a terminal symbol. For example, let the grammar be 〈V, Σ, S, P〉 with: V = {A,B,C,D,E,F,G,H,P,Q,R,S,T,U,V,W,X,Y,Z,p,q,r,s,t,u,v,w,x,y,z} Σ = {p,q,r,s,t,u,v,w,x,y,z} S = H P = { H → ABC A → (D A E A F) | P B → (X G Y)|Q C → Z D → W E → V F → U G → (TBS)R P → p Q → q R → r S → s T → t U → u V → v Y → y Z → z } A typical parse tree T for this grammar is in Figure 5. Upon editing, the parse tree T corresponding to grammar G becomes parse tree T', an ordered, labelled parse tree, having nodes labelled with elements of V, the root node of parse tree T' being labelled with S. An edited structured document can have a corresponding parse tree which is "gapped". A gapped parse tree can be in the form of a horizontally gapped tree or a vertically gapped tree. Figure 6 illustrates an example of a horizontal gapped tree, the gap being created with the deletion of the entire subtree rooted in the topmost "A". Figure 7 illustrates a vertically gapped tree, the vertical gap being created with the deletion of the topmost "A" and putting its children in its place. The ordered, labelled tree T' is a gapped tree of G if and only if there is a one-to-one mapping of the subtrees of T' into subtrees of a T such that any subtree of T' is mapped to a subtree of T having the same label on its root, and the mapping preserves the horizontal and vertical ordering of nodes as the nodes appeared in tree T' and T is a valid parse tree of G. A piecewise-structured editor for grammar G enables the transformation of one gapped tree of G into another such tree. While a gapped tree may not be a complete parse tree of G, the gapped tree can be made a complete parse tree purely by the insertion of new nodes and arcs in the tree. Hence, various "gaps" in T' can be "filled" to yield a valid parse tree T. Thus, the editing is piecewise, as the user is permitted to create any gapped tree T' via insertions/deletions of arcs and nodes. In order to make a piecewise-structured editing plausible, it must be efficiently determinable that a parse tree T' is a gapped tree of G. A method in accordance with the present invention enables this determination to be made. For example, given the example grammar, the complete parse tree illustrated in Fig. 5 could result. The following definitions are provided as background to the present invention. Definition 1 A gap grammar is a context-free grammar G = 〈V, Σ, S, P〉 such that for every A contained in V -Σ(nonterminal symbols), A → empty is contained in P, where empty (ε) is the empty string. This captures the idea of a horizontally-gapped tree. Thus, for the preceeding example grammar, the following productions must be added to make it a gap grammar: H → ε A → ε B → ε C → ε D → ε E → ε F → ε G → ε P → ε Q → ε R → ε T → ε U → ε V → ε W → ε X → ε Y → ε Z → ε Definition 2 axb → adb if there is a production x → y in G having d as a possible string in the language defined by the regular expression y. a → * b if either a = b or there is a → c such that a → c and c → * b. Definition 3 The reachability graph of the context-free grammar G = 〈V, Σ, S, P〉 is a directed graph whose set of nodes is V and whose arcs are the set of all ordered pairs 〈x, y〉 such that x,y are in V and x → ayb is in P. For symbols x,y in V, y is said to be reachable from x if there is a directed path from x to y of one or more arcs. This definition defines whether one node (e.g., y) is reachable from another node (e.g., x). The reachability graph for the example grammar is shown in Fig. 8. TABLE OF REACHABILITY FOR EXAMPLE GRAMMAR: H can reach A B C D E F G P p Q q R r S s T t U u V v W w X x Y y Z z A can reach A D E F P p U u V v W w B can reach B G Q q R r S s T t X x Y y C can reach Z z D can reach W w E can reach V v F can reach U u G can reach B G Q q R r S s T t X x Y y P can reach p Q can reach q R can reach r S can reach s T can reach t U can reach u V can reach v W can reach w X can reach x Y can reach y Z can reach z Definition 4 A strongly connected component of the reachability graph of G = 〈V, Σ, S, P〉 is a set of nodes in the reachability graph of G with the property that all nodes in the set are reachable from all other nodes of the set, and no node of the set is reachable from any node not in the set that is reachable from some node in the set. Consider the partition of the reachability graph of G = 〈V, Σ, S, P〉 into strongly connected components as illustrated in Fig. 8, for the example grammar. In Fig. 8, a set of regions are illustrated wherein a node located in one region goes to a node in another region when the two nodes are connected by an arrow. A strongly connected component having no outgoing (labelled) arcs has rank 0. A strongly connected component U having outgoing arcs has a rank one greater than that of the maximum of the ranks of the strongly connected components to which U is connected by outgoing arcs (from U). The rank of x in V is the rank of the strongly connected component in which x is found. Fig. 8 also labels the ranks of the strongly connected components. Definition 5 A symbol x in V of a context-free grammar G = 〈V, Σ, S, P〉 is recursive if and only if it is reachable from itself. A recursive symbol S is multiply-recursive if there exists a production in P from a symbol R in the strongly connected component containing S that can produce more than one instance of any symbol in the strongly connected component containing 5. A recursive symbol that is not multiply-recursive is singly-recursive. In the example grammar, A is multiply recursive and B and G are singly-recursive. All other symbols are not recursive. Definition 6 Let x be a symbol of the context-free grammar G = 〈V, Σ, S, P〉 We associate with every symbol x a regular expression called gap(x). We inductively define the regular expression gap(x) as follows: 1. If x in Σ is a terminal symbol (of rank 0), gap(x) = (x). 2. If x in V - Σ is a non-recursive symbol of rank 0, gap(x) = (ε|x). 3. If x in V is a non-recursive symbol of nonzero rank, gap(x) = (ε|x|R1|R2 ...) where R1, R2 ... are the right-hand-sides of productions for x with gap(w) substituted for each symbol w in the production. 4. If x in V is a multiply-recursive symbol, gap(x) = (x|S1|S2 ...)* where S1, S2 ... is the complete set of symbols reachable from x. 5. Let x in V be a singly-recursive symbol of G. Construct the sets of regular expressions L, M, and R as follows: Define SPLIT(exp) as: If SPLIT(exp) has already been evaluated for the same exp then return. Otherwise, if exp does not contain any symbol in the same strongly connected component as x then add exp with gap(w) substituted for each symbol w in exp to the set M. Otherwise, if exp is a symbol, then it must be in the same strongly connected component as x so do nothing. Otherwise, if the top-level operator in exp is * (or STAR), then there is an error as x could not possibly be singly-recursive. Otherwise, if the top-level operator in exp is |(or OR), then call SPLIT(alt) where alt is each of the alternatives of |(or OR) in turn. Otherwise, if the top-level operator in exp is CONCATENATE (juxtaposition), then there must exactly one term of the CONCATENATE that contains a symbol that is in the strongly connected component containing x. Add all of the symbols in terms to the left of this term to the set L; Add all of the symbols in terms to the right of this term to the set R; Call SPLIT(term) with the term containing a symbol that is in the strongly connected component containing x. Otherwise, if the top level operator is + (or PLUS), then there is an error as x could not possibly be singly-recursive. Otherwise, if the top level operator is ? (or OPTIONAL), rewrite the expression using OR and EMPTY (e.g., x? rewrites to (ε|x)). Then call SPLIT on the resulting expression. End of definition for SPLIT. Initialize L, M, and R to be empty sets; Call SPLIT(RHS) for every production for x in G where RHS is the right hand side of the production; Then left(x) = (L1|L2| ...) where L1, L2 ... is the set of all symbols appearing in the set L or reachable from a symbol in the set L; middle(x) = (ε|S1|S2 ... |M1|M2|...) where S1, S2 ... are the symbols in the strongly connected component containing x, M1, M2 is the set of regular expressions in the set M; right(x) = (R1 | R2 | ...) where R1, R2 ... is the set of all symbols appearing in the set R or reachable from a symbol in the set R. Then gap(x) = (left(x))* middle(x) (right(x))*. For the example grammar, SPLIT(B) will construct the following sets: L = {T,X} M = {gap(Q),gap(R)} R = {s,v} and left(B) = (T|t|X|x) middle(B) = (ε|B|G|gap(Q)|gap(R)) = (ε|B|G|Q|q|R|r) right(B) = (S|s|Y|y) The value of gap(x) for all of the nonterminals in the example grammar are: gap(H) = (ε|H|gap(A)gap(B)gap(C)) gap(A) = (A|D|E|F|P|U|V|W|p|u|v|w)* gap(B) = ((T|t|X|x)*(ε|B|G|gap(Q)|gap(R))(S|s|Y|y)*) gap(C) = (ε|C|gap(Z)) gap(D) = (ε|D|gap(W)) gap(E) = (ε|E|gap(V)) gap(F) = (ε|F|gap(U)) gap(G) = ((T|t|X|x)*(ε|B|G|gap(Q)|gap(R))(S|s|Y|y)*) gap(P) = (ε|P|gap(p)) gap(Q) = (ε|Q|gap(q)) gap(R) = (ε|R|gap(r)) gap(S) = (ε|S|gap(s)) gap(T)= (ε|T|gap(t)) gap(U) = (ε|U|gap(u)) gap(V) = (ε|V|gap(v)) gap(W) = (ε|W|gap(w)) gap(X) = (ε|X|gap(x)) gap(Y) = (ε|Y|gap(y)) gap(Z) = (ε|Z|gap(z)) The value of gap(x) expanded and simplified: gap(H) = (ε|H|(A|D|E|F|P|U|V|W|p|u|v|w)*(T|t|X|x)* (ε|B|G|Q|q|R|r)(S|s|Y|y)*(ε|C|Z|z)) gap(A) = (A|D|E|F|P|U|V|W|p|u|v|w)* gap(B)= (T|t|X|x)*(ε|B|G|Q|q|R|r)(S|s|Y|y)* gap(C) = ε|C|Z|z gap(D) = ε|D|W|w gap(E) = ε|E|V|v gap(F) = ε|F|U|u gap(G) = (T|t|X|x)*(ε|B|G|Q|q|R|r)(S|s|Y|y)* gap(P) = ε|P|p gap(Q) = ε|Q|q gap(R) = ε|R|r gap(S) = ε|S|s gap(T) = ε|T|t gap(U) = ε|U|u gap(V) = ε|V|v gap(W) = ε|W|w gap(X) = ε|X|x gap(Y) = ε|Y|y gap(Z) = ε|Z|z As is well-known, a finite-state machine can be constructed for any regular expression that efficiently checks whether or not a given string is a member of the language defined by the regular expression [see paper by R. McNaughton and H. Yamada, "Regular Expressions and State Graphs for Automata" in Sequential Machines: Selected Papers, Addison Wesley, 1964]. An exemplary finite state machine corresponding to gap(H) is shown in Fig. 9, gap(A) in Fig. 10, gap(B) in Fig. 11, gap(C) in Fig. 12, and gap(S) in Fig. 13. In these diagrams, accept states are shown as double circles while fail states are shown as single circles. In accordance with the present invention, the editor enables determination of whether tree T' is valid. In a first embodiment, the validity of the entire tree is checked by the editor in accordance with the following: Let R be a regular expression. It is well-known that the strings of a language defined by a regular expression can be recognized in deterministic time of the order of the string length. Thus, given a regular expression R, a procedure recognize(R) is compiled which is parameterized by strings (ordered lists of symbols in V) S that return true if and only if S is in R where recognize(R) runs in deterministic time of order length(S). By virtue of the previous theorem, the deterministic procedure below returns true if and only if T' is a gapped tree of G. Moreover, it runs in time linear in the size of the tree (gapped or not). For each node labelled A in T' with immediate descendant nodes labelled B1,...,Bk do If recognize(A)(B 1 ,...,B k ) then true else return(false) Return(true) A may have no immediate descendants, e.g., B1,...,Bk = ε. The finite state machines constructed by methods in accordance with the present invention determine whether or not the sub-structures of a document fed them as input can comprise an instance of a parse tree of a gap grammar. The names of the children of each and every nonterminal node (in order, as a string) must be accepted by the finite state machine named by the parent node for the tree to be an instance of a parse tree of a gap grammar. If this condition holds, then the parse tree is that of a weakly valid document, otherwise, it is the parse tree of neither a valid nor weakly valid document. In another embodiment of the invention, the editor needs only check certain subtrees of tree T' created by the insertion or deletion of nodes and arcs in tree T. In this embodiment, for an inserted node, only the parent of a newly inserted node with its new children and the newly inserted node and its children are checked. For a deleted node, only the parent of a newly deleted node with its new children are checked. This alternative procedure is performed in accordance with the following: parent(node) takes a node as a parameter and returns its parent node in the tree T'; childnames(node) takes a node as a parameter and returns the ordered list of child labels below the node in the tree T'; fsm(node) takes a node as a parameter and returns the finite-state machine (regular expression) corresponding to the label of that node in the tree T'; For each new node n in T' do If not(recognize(fsm(parent(n)))(childnames(parent(n)))) then return false If not(recognize(fsm(n))(childnames(n))) then return false For each parent node n (that still exists) of a deleted node in T' do If not(recognize(fsm(n))(childnames(n))) then return false Return true. The latter procedure runs in time proportional to the sum of the number of nodes below the old parents of inserted nodes and the number of nodes below the parents of deleted nodes. For example, to check if the nodes below the node "H" of the tree in Figure 7 are allowed in the parse tree of a weakly valid document, the labels of the children "DAEAFBC" are formed into a string and attempted to be recognized by the finite state machine labelled by H in Figure 9. Starting in state 1 the machine will make the following transitions for the given string: state 1 scans D resulting in state 2 scans A resulting in state 2 scans E resulting in state 2 scans A resulting in state 2 scans F resulting in state 2 scans B resulting in state 4 scans C resulting in state 5. Since state 5 is marked as an accept state (double circle) the sequence of children is possible in a parse tree of a weakly valid document. Accordingly, the editor/parser enables efficient determination of whether a given labelled, ordered tree can be extended to be a parse tree of given context-free grammar. The editor/parser can be a programmed computer which determines whether a given parse tree can be extended to a valid parse tree. The determination can be made either in time linear to the number of nodes in the newly created tree or can be modified to make the determination in time linear to the size of only the changed portion of the parse tree. Although methods are described above in conjunction with the SGML grammar-based specification, the present invention could be applied to any document representation described by a formal context-free grammar-based specification, such as open document architecture (ODA).

Claims: A method of editing a document, comprising determining whether a given parse tree (fig. 5) generated from said document is a weakly-valid parse tree of said document according to a given context-free grammar, a weakly valid parse tree being a parse tree that can be extended into a valid parse tree that satisfies the given context-free grammar with only an insertion of additional nodes, characterized in that it includes the steps of: constructing predetermined regular expressions (figs. 9-13) for selected nodes of the parse tree according to said given context-free grammar; checking for each selected node of the given parse tree whether the names of the children of said node form a string in a language defined by the corresponding predetermined regular expression; and determining if the given parse tree is a weakly-valid parse tree by verifying that said checking step succeeds for all selected nodes. A method according to claim 1, wherein the predetermined regular expression attached to a node labeled by a symbol x is an expression "gap(x)" defined by : if x is a terminal symbol of the grammar, gap(x) = (x); if x is a non recursive non terminal symbol of rank zero, gap(x) = (ε|x); if x is a non-recursive non terminal symbol, gap(x) = (ε |x|R1|R2 ...) where R1, R2, are the right-hand-sides of productions for x with gap(w) substituted for each symbol w in the production; if x is a multiply-recursive non terminal symbol, gap(x) = (ε|x|S1|S2 ...) where S1, S2 is the complete set of symbols reachable from x; if x is a singly-recursive non terminal symbol, gap(x) = (left(x)*middle(x)(right(x))*, where: SPLIT(exp) is defined as: if exp does not contain any symbol in the same strongly connected component as x than add exp with gap(w) substituted for each symbol w in exp to the set M, otherwise, if the top-level operator in exp is "OR", then call SPLIT(alt) where alt is each of the alternatives of"OR" in turn, otherwise, if the top-level operator in exp is CONCATENATE (juxtaposition), then there must be exactly one term of the CONCATENATE that contains a symbol that is the strongly connected component containing x, add all the symbols in terms to the left of this term to the set L; add all the symbols in terms to the right of this term to the set R, call SPLIT(term) with the term containing a symbol that is in the strongly connected component containing x, otherwise if the top-level operator is "?" (OPTIONAL), rewrite the expression using OR and EMPTY, then call SPLIT on the resulting expression; left(x) = (L1|L2| ...), where L1, L2 is the set of all symbols appearing in the set L or reachable from a symbol in the set L, middle(x) = (ε|x|T1|T2 ... |M1|M2 ...), where T1, T2 are the symbols in the strongly connected component containing x and M1, M2 is the set of regular expressions in the set M, right(x) = (U1|U2 ...) where U1, U2 is the set of all symbols appearing in the set R or reachable from the set R. A method according to any one of claims 1 to 2, wherein the selected nodes are the nodes labeled with nonterminal symbol of the given context-free grammar. A method according to any one of claims 1 to 3, wherein the selected nodes are only the root nodes of certain subtrees of the parse tree. A method of editing a document according to a computer language, characterized in that it comprises determining whether a given parse tree (fig. 5) generated from the document can be extended to a valid parse tree of a given context-free grammar G with only an insertion of additional nodes, by performing the steps of: constructing regular expression gap(x) for each nonterminal symbol x of the given context-free grammar G; checking each node having one of the non-terminal symbols of the given parse tree for child nodes whose names form a string in a language of the corresponding regular expression gap(x); and determining, when the names form a string in the language of the corresponding regular expression gap(x), that the given parse tree can be extended to the valid parse tree. A method according to claim 5, including converting the regular expressions to finite state machines.

Inventors: BROWN ALLEN L JR, MARSHALL SIDNEY W, BROWN,ALLEN L. JR., MARSHALL, SIDNEY W.
Applicants: XEROX CORP, XEROX CORPORATION
Application year: 1994
Main classification: G06F17
IPC codes: G06F17